MATHEMATICS 184 (Section 922) - TERM EXAM 3
NAME:
STUDENT ID NUMBER:
SIGNATURE:
INSTRUCTIONS: No notes or books are to be used. Calculators
are allowed. No credit will be given for the correct answer without
the (correct) accompanying work. Use the back of the pages if you
need extra space.
1.(a) Find the slope of the graph of f (x) = 1 − ex at the point
where it crosses the x-axis.
[2]
(b) Find the equation of the tangent line to the curve at this point.
[2]
(c) Find the equation of the line which is perpendicular to the tangent line at this point. (This line is known as the normal line.)
[3]
2. The area A, in cm2 of my fishtank that is covered in algae after
t days grows according to the equation:
A(t) = 0.1(1.05)t
(a) After 1 week (7 days), how much of the surface of the tank is
covered (in cm2 )?
[2]
(b) If my fishtank has the dimensions:
[4]
After how many days is the entire tank covered?
(c) After 1 week (7 days), what is the growth rate (instantaneous)
of the algae? Provide units with your answer.
[4]
3. Using the definition of the derivative (ie. no shortcuts), find the
derivative function f 0 (x) for the function
x2 + 3
f (x) =
x−1
[10]
4. Differentiate the following functions:
(a) f (x) = x2 2x
(b) y =
x3 +7x
(3x+2)6
(c) g(x) =
x3 +2x2 +3x
x−1
[3]
[4]
[3]
t2
(d) f (t) = 2e
[4]
(e) z = 7cos(7x) − sin2 (t2 )
[4]